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State what the two numbers represent
Suppose qa and qb are quantiles of the same declared normal loss law at distinct confidence levels a and b. They must share the same unit, horizon and loss definition. Two values from incompatible models cannot identify one parameter pair.
Subtract the quantile equations
za and zb are distinct standard normal quantiles; the resulting scale must be nonnegative.
Check an internally generated example
Original quantiles 16.448536... at 95% and 23.263479... at 99% come from μL=0 and σL=10. Using full precision recovers dispersion 10 and mean zero. Rounded published values can produce a small reconstruction difference.
Keep identification separate from evidence
Solving two equations does not test normality, calibrate an empirical distribution or validate the confidence interpretation. Other loss laws can share those two quantiles while having different tails.